#7196715

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45388	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1775 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

(1,2,4,3)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22)(23,31)(24,35)(25,30)(26,32)(27,28)(29,38)(33,45)(34,46)(36,51)(37,42)(39,43)(40,52)(41,48)(44,50)(47,49);(2,3)(7,13)(8,16)(10,17)(11,15)(21,23,26,27)(22,28,32,31)(24,33,37,38)(25,39,41,34)(29,42,45,35)(30,46,48,43)(36,50,40,49)(44,51,47,52);(5,7,9,13)(6,10,12,17)(8,14,16,19)(11,18,15,20)(21,24,23,33,26,37,27,38)(22,29,28,42,32,45,31,35)(25,40,34,50,41,36,39,49)(30,47,43,51,48,44,46,52);(1,3,4,2)(7,13)(10,17)(14,19)(18,20)(21,25,26,41)(22,30,32,48)(23,34,27,39)(24,36,37,40)(28,43,31,46)(29,44,45,47)(33,49,38,50)(35,51,42,52);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(21,23,26,27)(22,31,32,28)(24,33,37,38)(25,39,41,34)(29,35,45,42)(30,43,48,46)(36,50,40,49)(44,52,47,51)

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,c^2=ghi,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7103911);